We devoted the rest of the paper to the study of the integrability properties of this equation.
The results of Sections –
can be seen as a nice example of the properties that were
found for the two classes of linearizable equations belonging
to Boll's classifications, namely the trapezoidal \(H^{4}\) 
equations and the \(H^{6}\)  equations [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}.
Indeed we proved in Section  that the
non-autonomous Hietarinta equation can be mapped into a
particular case of the \(_{1}D_{4}\)  equation, an equation
belonging to the \(H^{6}\)  class.
We remark that the identification of the non-autonomous
Hietarinta equation (REF ) with equation (REF )
is possible only in the extended framework of
the consistency around the cube given in [8]}, [9]}, [10]}, [11]},
since in order to transform the non-autonomous
Hietarinta equation (REF ) into (REF )
the non-autonomous Möbius transformation (REF )
must be used.
In the framework of [12]} this is not possible
since transformations of this kind are not allowed.
Therefore the construction carried out in this paper
is “necessary” to obtain this result.
